On previous subsections we analyzed the stability of Fourier method
in terms of two main ingredients: weighted 
-stability on the one hand,
and high frequencies instability on the other hand.
Here we would like to show how both of these
ingredients contribute to the actual
performance of the Fourier method.
We first address the issue of resolution.
We were left with the impression that the weak -instability
is a rather 'rare occurrence', as it is excited only
in the presence of nonsmooth initial data. But in fact, the mechanism of
this weak -instability will be excited whenever the Fourier method
lacks enough resolution.
In this context let us first note
that the solution of the underlying hyperbolic problem may develop
large spatial gradients due to the almost impinging characteristics along
the zeroes of the increasing part of a(x).
Consequently, the Fourier method might not have enough modes to
resolve these large gradients as they grow in time.
This tells us that independent whether the initial data are smooth
or not, the computed approximation will then 'see' the underlying
solution as a nonsmooth one, and this lack of
resolution will be recorded by a slower decay of the computed
Fourier modes. The latter will experience
the high-frequency instability discussed earlier and
this in turn will lead to the linear -growth. Our prototype
example of is case in point: according to Corollary
3.2,
one needs here at least modes in order to resolve
the solution, for otherwise, (weak-in.19) shows that spurious
oscillations will contaminate the whole computed spectrum.
We conclude that the lack of resolution manifests itself
as a weak -instability. This phenomenon is demonstrated
in Figures 3.5-3.9,
describing the Fourier
method (weighted.6) subject to (the perfectly smooth ...) initial condition,
. Figure 3.5 shows how the Fourier method with fixed
number of N=64 modes propagates information regarding
the steepening of the Fourier solution in physical
space, from low modes to the
high ones.
And, as this information is being transferred to the high
modes, their amplification become more noticeable
as time progresses in Figures 3.5a-3.5d.
Consequently, though N=64 modes are sufficient to resolve
the exact solution at , Figure 3.6c-d shows that
at later time, t=3 and in particular t=5, the under resolved
Fourier solution with 64-modes will be completely dominated by the
spurious centered spike. This loss of resolution requires more
modes as time progresses. Figure 3.7 shows how the Fourier
method is able to resolve the exact solution at t=3.5, once
'sufficiently many' modes, are used, in agreement
with Corollary 3.3. According to Figures 3.8 and 3.9,
modes are
required to correctly resolve the two strong boundary dipoles at t=4,
yet at t=8 the Fourier solution will be completely dominated by the
spurious centered spike.
Assuming that the Fourier method contains sufficiently many modes dictated by the requirement of resolution, we now turn to the second issue of this section concerning the convergence of the Fourier method.
. The requirement from the initial data to have at least -regularity is clearly necessary in order to make sense of its pointwise interpolant.